I don't know the answer to this question, but will make an extended remark.
Let $X$ be a finite set and let $f:X^n\to X$ be any $n$-ary operation on $X$, $n>0$.
Claim. The following conditions are equivalent.
(i) $f$ is surjective with uniform kernel.
(Equivalently, for each $a\in X$ the set
$f^{-1}(a)=\{(x_{1},\dots,x_{n}):f(x_{1},\dots,x_{n})=a\}$
has size $|X|^{n-1}.$)
(ii) There exist $n$-ary operations on $X$, $T_2,\ldots, T_n$ and $S_1,\ldots, S_n$ such that, if $G,H$ are $$G(\bar{x}) = (f(\bar{x}), T_2(\bar{x}), \ldots, T_n(\bar{x}))$$ and $$H(\bar{x}) = (S_1(\bar{x}), S_2(\bar{x}), \ldots, S_n(\bar{x})),$$ then $G$ and $H$ are inverse bijections between $X^n$ and $X^n$.
The question asks, if $V$ is a variety satisfying:
I. $V$ is finitely axiomatizable.
II. $V$ is generated by its finite members.
III. Item (i) above holds for the interpretation
of any fundamental operation of arity at least $1$
on each finite member of $V$,
then must Item (ii) above hold in the strong sense that the $S$'s and $T$'s are term operations, but in the weak sense that we allow other parameters $m$ and $r$ in place of some instances of $n$?
Roughly, this asks if having Item (i) hold throughout the finite part of $V$ implies that Item (ii) is enforced by the equational theory of $V$.
This seems plausible to me, but it also seems that there are some extraneous elements in the question. I don't think that $V$ being finitely axiomatizable is relevant. I don't think the additional flexibility of introducing parameters $m$ and $r$ possibly different from $n$ helps, but I haven't tried to check any examples. (It is clear at least that $m$ must equal $r$ if $V$ is generated by its finite members.) I also think the result, if true, is not a property of varieties; that is, the question can be asked for a single (fundamental) operation of $V$: if $V$ is generated by its finite members and $f$ is a fundamental operation of positive arity satisfying Item (i) above, then must Item (ii) above hold?
Here is a sketch of a proof of the claim.
(ii) implies (i): Let $\pi_1: X^n\to X$ be the first projection map. It is surjective with uniform kernel. Since $G: X^n\to X^n$ is a bijection, $\pi_1\circ G$ ( = $f$) is also surjective with uniform kernel.
(i) implies (ii): For each $a\in X$, choose a bijection $\beta_a: f^{-1}(a)\to X^{n-1}$. (Item (i) is the statement that such a bijection exists.) If $$G: X^n\to X^n: \bar{x}\mapsto (f(\bar{x}),\beta_{f(\bar{x})}(\bar{x})),$$ then $G$ is a bijection, and the appropriate component functions exist for both $G$ and its inverse $H$.